Fourier Definition and 1000 Threads

Is there some properties I should be aware of? after making the relevant substitutions, I ended up with $2 = 1 + \sum\nolimits_{m=0}^\infty \frac{4}{(2m+1)\pi}\sin(\frac{(2m+1)\pi}{2})$ but I can't get past this

I've just started learning Fourier series and I'm having trouble understanding it. What do they actually do? And what does the amplitude-frequency show me? I'm asking as a rookie in signal analysis, so if you could explain it to me as simple as you can it will be of great help. Thanks!

Sorry if I am posting in the wrong place. I'm really interested in the Fourier series, but I'm not an expert on it yet. I am very well aware yoy can do it with sound waves, but can you manipulate any other waves? What about light waves? And for absorption, how can you measure the...

Hi all, Only few days back I got the idea of probability density function. (Till that day , I believed that pdf plot shows the probability. Now I know why it is density function.) Now I have a doubt on CTFT (continuous time Fourier transform). This is a concept I got from my...

I've been assigned the following homework: I have to compute the spectral density of a QFT and in order to do so I have to compute Fourier tranform of the following quantity (in Minkowsky signature, mostly minus) \rho\left(p\right) = \int \frac{1}{\left(-x^2 + i \epsilon...

Somehow I have really hard time wrapping my head around the concept.I mean,I get it,but I can't seem to solve any questions regarding it. Here are some examples ,and I just get stuck.Its a part of test,so I think it shouldn't be that hard to solve,and if it looks hard,I know there are some...

Homework Statement Sawtooth signal with To = 1, at T=0, x = 0, at T=1, x =1 verify: a_{k} = \left\{\begin{matrix} \frac{1}{2}, for k=0; & \\\frac{j}{2\pi k}, for k \neq 0; & \end{matrix}\right. Homework Equations \frac{1}{T_{0}} \int_{0}^{T_{0}} te^{-j(2\pi/T_{0}))kt}dt The Attempt...

Homework Statement An oscillator with free period \tau is critically damped and subjected to a force with the saw-tooth form \F(t)=c(t-n\tau) for (n-0.5)\tau<t<(n+0.5)\tau for each integer n. Find the amplitudes a_n of oscillation at the angular frequencies 2\pi n/\tau if c is a...

I'm having trouble finding a definite answer to this question: When finding the Fourier series of a function is it always possible to find ##a_0## by first finding ##a_n## and just plugging in ##n=0##?

Homework Statement The problem: Justify the following equalities: \cot x = i\coth (ix) = i \sum^\infty_{n=-\infty} \frac{ix}{(ix)^2+(n\pi)^2}=\sum^\infty_{n=-\infty}\frac{x}{x^2+(n\pi)^2} I am trying to figure out how to start this. When I insert the Euler identity of \coth (using...

Hi all my first post as I need to seek help! I have just learned some simple Fourier series stuff and would like to be able to plot my answers in matlab. Assuming this is correct I was wondering if someone would be able to walk me through plotting this equation in Matlab...

Hi, I'm following the proof of the "Scaling Property of the Fourier Transform" from here: http://www.thefouriertransform.com/transform/properties.php ...but don't understand how they went from the integral to the right hand term here: The definition of the Fourier Trasform they...

Solving a "simple" second order PDE, do I need the Fourier? Homework Statement The problem as given: y'' + 2y' + 5y = 10\cos t We want to find the general solution and the steady-state solution. We're using \mu y'' + c y' + k y = F(t) as our general form. OK, so I first want the general...

Consider \(u_t = -u_{nxxx} - 3(u^2)_{nx}\). The Fourier Transform is linear so taking the Inverse Fourier transform of the Fourier Transform on the RHS we have \begin{align} -\mathcal{F}^{-1}\left[\mathcal{F}\left[u_{nxxx} - 3(u^2)_{nx}\right]\right] &= -\mathcal{F}^{-1}...

3. Fourier sin series for f(x) = 1, 0 < x < pi is given by 1 = 4/n E 1/ (2n-1) times sin (2n-1) x, (0 < x < n). Using this, find the Fourier sinc series for f(x)= 1, on 0 < x < c where c > 0. Then find the Fourier series for g(x), x > 0 where g(x) = 1, 0 < x < c, -1, c < x < 2c, g (x + 2c)...

2. Fourier cosine series correspondence for f(x)= x, o < x < pi given by x ~ pi / 2 - 4/n, E infinity on top and n=1 on bottom. cos (an-1)/x / (2n-1)squared, (0 < x < pi). Explain why this correspondence is actually an equality for 0 is less than or equal to x and x is less than or equal to...

When a Fourier series contains only sine and cosine terms, evaluating the series isn't too difficult. However, I want to show a Fourier series with sine and sinh converges to \(\frac{\pi}{16}\). \[ T(50, 50) = \sum_{n = 1}^{\infty}...

Today I found a program, which does Fourier transforms on pictures and tried it on some basic patterns. One of those was a lattice of dots and I have attached this and its Fourier transform to the thread. I would very much like if someone in basic details could explain what is going on. Why...

Homework Statement The problem/question is attached in the file called "homework". In the third signal (the peridic rectangular wave), I am requested (sub-question b) to find the Fourier series of the wave. Homework Equations The file called "solution" presents a detailed solution to the...

Finite Fourier Transform on a 2d wave How does the finite Fourier transform work exactly? The transform of f(x) is \widetilde{f}(\lambda_{n}) =\int^{L}_{0} f(x) X_{n} dx If I had a 3d wave equation pde and I applied Finite Fourier transform on the pde for z(x,y,t)=X(x)Y(y)T(t)...

Is this college or graduate math? Is it pure or applied math? Is it useful for physics and electrical engineering?

Odd & Even Functions (was thread "Fourier Series") Homework Statement determine if the functions below are odd even or neither: a) f(x)=x^2+2 b) f(x)=(x^2+2)tan(x^2) c) f(x) = (x^2+2)sin(x)tan(x^2)Homework Equations even - f(x) = f(-x) odd - f(-x)=-f(x) The Attempt at a Solution I've managed...

Hi all, as a physics student, I seldom use Fourier transform but from my understanding, given a periodic function you can decompose the function into sine function with different frequencies. Also, to get a ultra short pulse in time domain, this would require mixing many frequencies. I would...

Homework Statement Determine the Fourier series for the periodic function of period 2∏ defined by: -2 when (-∏ ) ∠ x ∠ (-∏/2) f(x)= 2 when ( -∏/2) ∠ x ∠ (∏ /2) -2 when (∏/2) ∠ x ∠ (∏) how to start i?. I have already drawn it but what next. thank you...

Hey there! I'm trying to calculate the Fourier Series for sin2x on [-π, π] For a0 I found 1/2. (By determining the average value of the function on the interval) Since sin2x is even, I know that bn = 0. Now, for an.. The following link shows the integral I used to try to evaluate an...

Homework Statement \psi (x) = Ne^{ \frac{-|x|}{a}+ \frac{ixp_o}{/hbar}} Compute Fourier transform defined by ##\phi (p) = \frac{1}{ \sqrt{2 \pi \hbar}} \int \psi (x) e^{ \frac{-ipx} {\hbar}} dx## to obtain ## \phi (x) ## Homework Equations Fourier transform = ##g(x)= \frac {1}{2 \pi} \int...

Square Pulse Train Fourier Series help?? Homework Statement problem+directions below: Homework Equations ω=2\pif β=\frac{2\pi}{\lambda} The Attempt at a Solution Since the problem asks to make all time-dependent sinusoidal functions deal with x-direction, i don't think i need to...

Hi, I have the following question: A signal x(t) which is band-limited to 10kHz is sampled with a sampling frequency of 20kHz. The DFT (Discrete Fourier Transform) of N= 1000 samples of x(n) is then computed. To what analogue frequency does the index k=120 respond to? I'm trying to...

Hi all, I want to calculate \int_0^{\infty}e^{-a t^2}\cos(2xt)dt=\frac{1}{2}\sqrt{\frac{\pi}{a}}e^{\frac{-x^2}{a}}. The answer is known from the literature, but I don't know how to do it step by step. Any one has a clue? Thanks. Jo

Homework Statement Wow LaTex fail... anyone know how i make this look like not ****? I had it looking all pretty on http://www.codecogs.com/latex/eqneditor.php and it gave me these codes as my latex markup but it didn't come out right... why do they look like source code and not the pretty...

Homework Statement Let \hat{u}_k the Fourier coefficients of 2-periodic function u(t)=t with t\in [0,2). Evaluate the sum of the serie: \sum_{k=-\infty}^{\infty}\hat{u}_k e^{\pi i k t} for t= 2 Ok, I think there is a trick that I don't know... \sum_{k=-\infty}^{\infty}\hat{u}_k...

Homework Statement Let u(t)=2-\cos(t)+\sin(2t)- \cos(3t)+ \sin(4t) Evaluate: \int_0^{2\pi}u^2(t)\mbox{d}t Homework Equations The Attempt at a Solution Sorry, I don't have any idea :(... As I can see \int_0^{2\pi}u^2(t)\mbox{d}t is similar to the first term of...

I took f(t) = SIN(10*t) +SIN(5*t) and got this f(0) = 0 f(1) = -1.5 f(2) = 0.4 f(3) = -0.3 now I tried to do the DFT Fs = 4Hz N = 4 samples 3 f[r] = Ʃ x[k]ε^(-j(2πkr/4) k=0 f[r] = 0 -1.5ε^(-j(2πr/4) + 0.4ε^(-j(2π(2)r/4) -0.3ε^(-j(2π(3)r/4) f[0] = 0 - 1.5 +...

Hey. I'm looking for a proof of: Theorem: If f \in C^1(\mathbb{T}), then the Fourier series converges to f uniformly (and hence also pointwise.) I have looked around for it, googled, etc, but I only found proofs which used theorem they did not prove. (Or I misunderstood what they said.) I'd...

Hi there, I am trying to get some practice with Fourier Transforms, there is a long way to go. For example, let me consider the function $$ \gamma (t) = \int_{-\infty}^{t} C(t-\tau) \sigma(\tau) \mathrm{d}{\tau}$$ Defining the Fourier Transform as $$ \gamma(\omega) = \frac{1}{2 \pi}...

I was looking through some examples which applied the duality principle while studying for an up and coming exam when it hit me that the transform applied 4 times gives you back the same function. So is there some theory that uses this? perhaps some sort of operator? I thought it...

Homework Statement Compute the Fourier transform of a function of norm f(\norm{x}). Homework Equations \mathbb{F}{\frac{1}{1+\norm{x}} The Attempt at a Solution Attempt at using Cauchy theorem and the contour integral with the contour [(-R,R),(R,R+ip),(R+ip,-R+ip),(-R+ip,-R)] does...

Everything is in the picture I've attached. I believe my work is right because it's not that difficult of a problem but what I'm having a hard time seeing is how i go from the coefficient bn that I've calculated to the final solution. Maybe I did screw up or maybe there's an identity I'm not...

greetings, Can anyone tell me when we should use Laplace transform and Fourier transform? It seems both of them are equal except σ . thanks in advanced.

Homework Statement Fourier coefficients: A_0=0, a_n=0, b_n=2/(n∏) ; period p=2 Homework Equations Fourier series The Attempt at a Solution Attempt was not good enough!

Hey everyone, I've been reading up a bit on control systems theory, and needed to brush up a bit on my Laplace transforms. I know how to transform and invert the transform for pretty much every reasonable function, I don't have any technical issue with that. My only problem is that some...

Checked around a buch and could not find any help. But I needed help with: Understanding that if I get the Inverse FT of K-space data, what is the scaling on the X-space (object space) resultant image/data i.e. for every tick on the axis, how do I know the spatial length? More detailed...

Hi, I wish to obtain the Fourier series of the signal in red (please see attached figure fig1_sine_plots.png). Basically, it is a full-wave rectified 3f sinusoid, where f = 50Hz. The blue signal represents a sinusoid with frequency f = 50Hz. In the following equations (please see attached...

Homework Statement (a) On (-π,π), find the Fourier series of f(x) = x. (b) Hence, or otherwise, find the Fourier series of g(x) = x2 (c) Hence, show that \sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90} Homework Equations f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos \frac{n...

Hello everyone: I have some question using the FFT in MATLAB for data interpolating. I don't know what the relation between the normal Fourier series and the real, image number. For example, given a set of measurement data, I can use the curve fitting toolbox to fit a curve. The general...

Hi! I was wondering how you would graph a Fourier series for a square wave that periodic. My textbook gave something proper that I would expect (just google 'Square Wave Fourier Series Expansion') while my TI - 84 gave a image like the one somewhere in this thread. It would be nice if anybody...

Homework Statement I am trying to figure out how to graph the signal spectra of an AM signal where the message m(t) is multiplied by the carrier, which is sin^3 (wt) instead of cos (wt). I can do the FT but I do not know how to graph this since there are imaginary numbers as coefficients...

Homework Statement Hi, this is not a homework question per se, but something I'm wondering. Let C be a circulant n x n matrix, let x, b, be vectors such that C x = b. We would like to find a solution x. One way is to use the DFT: According to section 5, In Linear Equations, in the wikipedia...

The Fourier series can be used to represent an arbitrary function within the interval from - π to + π even though function does not continue or repeat outside this interval. Outside this interval the Fourier series expression will repeat faithfully from period to period irrespective of whether...

The Fourier series can be used to represent an arbitrary function within the interval from - π to + π even though function does not continue or repeat outside this interval. Outside this interval the Fourier series expression will repeat faithfully from period to period irrespective of whether...